Howdy Gang! Okay let's see if I can toss a few more bits out there.
How far south are we talking (re difficulties with southerly travel)? Australia is only between 10 and 40 lat south (maybe 44 if you add Tasmania?). There's a lot more down there!!
Moon phases don't pose a problem. The Moon is a certain shape, and it probably has a rotation so that the growth that it induces can be regulated according to its design.
I'll risk a trigonometry-free argument, re the rope view of a lateral ship transit. What I like about that experiment is that it removes the normal refraction argument. The ship is mostly at pretty much an equal distance from the viewer the whole time (with some potential confusion, it seems to me, regarding what constitutes a truly straight and parallel line with the rope). Here is the segment, from ENAG.
What's interesting is that the curvature math suggests a height difference that would be greater than the ship, or a large fraction of the ship.
Now other thoughts.
For the most part, spherical-earth believers are just that: believers. They believe that the gravitation-based system, with its mathematical enormity, has been worked out successfully and sufficiently, so that it is a safe belief. I would wager that it's a rare spherical-earth believer who understands even the gravitational math that keeps the sun, earth, and moon in their supposed relationship.
The rotundity-through-rotation movement seems suspect. Liquid matter does not necessarily form a roundness. I would expect a molten blob to fail to cohere in a roundness. But this is essential to the creation-from-nothing-but-spinning-exploding-matter theory.
Air is plainly expansive. It is truly remarkable that it is held in place by gravity. I would expect it to just float away, especially given the opportunity based on the various motions that are available to it, according to astronomy.
Why doesn't light wrap around the earth and deny night-time to the other side? The sun is so large, and the earth is so small, and there should be a lot of refraction.
Now, for some religious blathering. Man's preference for sin explains his preference for imaginary systems. Astronomy is bizarre and imaginary, but, it is coherent insofar as it is God-denying, and Scripture-denying. A pattern is worthy of notice.
It is common to posit the 'days' of Genesis to be ages, but the sabbath was never understood that way. That reflects a pick-and-choose view of the Scriptures.
Feel free to ignore the religious points.
Gravitation belongs to metaphysics. This is a partly religious observation, partly a matter of philosophical classification.